Class 8: Multiple Sequence Alignment
Bioinformatics
Andrés Aravena
December 7, 2023
Multiple Sequence alignment
Source: Uçarlı, Cüneyt, Liam J. McGuffin, Süleyman Çaputlu, Andres Aravena, and Filiz Gürel. “Genetic Diversity at the Dhn3 Locus in Turkish_Hordeum Spontaneum_Populations with Comparative Structural Analyses.” Scientific Reports (2016) https://doi.org/10.1038/srep20966.
Why Multiple Sequence Alignment?
To study closely related genes or proteins
- to find conserved domains
To find the evolutionary relationships
- the base of phylogenetic trees
To identify shared patterns among related genes
- conserved sequences can be binding sites
In the previous chapter…
We discussed pairwise alignment
Global, using Needleman & Wunsch algorithm
- also for semi-global alignment
Local, using Smith & Waterman algorithm
Both methods build a dot-plot matrix
We build a matrix with \(m_1\) rows and \(m_2\) columns
We write the sequence \(s_1\) in the
rows,
and the sequence \(s_2\) in the
columns
The computational cost is \(O(m_1 m_2)\)
(\(m_1\) is the length of \(s_1\), \(m_2\) is the length of \(s_2\))
Filling the dot-plot matrix
To find the optimal alignment, we look for diagonals in the matrix that maximize the total score
Every cell \(M_{ij}\) in the matrix
has initially the value \[M_{ij}=\text{Score}_2(s_{1}[i],s_{2}[j])\]
where \(s_{1}[i]\) is the letter in
position \(i\) of sequence \(s_1\),
and \(s_{2}[j]\) is the letter in
position \(j\) of \(s_2\)
Pairwise to three-wise alignment
To aligning two sequences, we build a dot-plot matrix.
That is, a rectangle.
To align three sequences, we need a three-dimensional array.
That is, a cube.
Each cell \(M_{ijk}\) has value \[M_{ijk}=\text{Score}_3(s_{1}[i],s_{2}[j], s_{3}[k])\]
Then we find the diagonals
Usually, external gaps do not count, but internal gaps count
That is, these are semi-global alignments
Any path from a border to another border will be an alignment
We look for the optimal alignment
Cost of three-wise alignment
If the three sequences have length \(m_1, m_2,\) and \(m_3,\) then building the cube has cost \[O(m_1\cdot m_2\cdot m_3)\]
Always look for the big picture
To simplify, we assume that all sequences have length \(m\)
Then the cost of three-wise alignment is \[O(m^3)\]
Multiple alignment
Following the same idea…
To align \(N\) sequences, we need a dot-plot in \(N\) dimensions
\[M_{i_1,\ldots,i_N}=\text{Score}_N(s_{1}[i_1],s_{2}[i_2],…,s_{N}[i_N])\]
Therefore, if the average sequence length is \(m,\) then the cost is \[O(m^N)\]
How much is that?
To fix ideas, assume that \(m=1000\)
(That is a typical size for a bacterial gene)
The computational cost is \(O(1000^N)\)
In other words, the cost is \(O(10^{3N})\)
How many seconds
Now assume that the computer can do one million comparisons each second
The number of seconds is then \[O(10^{3N-6})\]
Exercise: How many seconds will it take for 2, 4, 8, and 12 sequences?
How much is that?
Under these hypothesis we have this table
| \(N\) | Seconds | In words |
|---|---|---|
| 2 | \(10^0\) | 1 sec |
| 4 | \(10^6\) | 1 million seconds |
| 8 | \(10^{18}\) | 1 trillion/quintillion seconds |
| 12 | \(10^{30}\) | a lot of time |
Exercise 1
Translate these numbers to days, years, etc.
(Approximate answer are OK. We only need one significant figure)
Exercise 2
How do these numbers change if \(m\) changes?
Exercise 3
What happens if the computers are 1000 times faster?
Exercise 4
What is the largest multiple alignment that you can do in your life?
Exercise 5
What is the largest number of sequences that can be aligned?
Exercise 6
What can we do to align more sequences?
Heuristic
This is clearly too expensive, so we need heuristics
(i.e. solving a similar but simpler problem)
One common idea is to do a progressive alignment
- start with a pairwise alignment,
- and then align the rest one by one
There are too many heuristics
There are several ways to simplify the original problem
Thus, there are many approximate solutions
The main differences are:
- How to decide what to align first?
- Can new sequences change the previous alignment?
- Can \(s_k\) change the alignment of \((s_1,…,s_{k-1})\)?
- How to use additional information (like 3D structure)?
- What is the formula for \(\text{Score}_k(s_{1}[i_1],…,s_{k}[i_k])\)?
Some common Multiple-Alignment tools
- Clustal
- ClustalW, ClustalX, Clustal Omega
- T-Coffee
- 3D-Coffee
- MUSCLE
- MAFFT
Clustal
Clustal was the first popular multiple sequence aligner
Versions: Clustal 1, Clustal 2, Clustal 3, Clustal 4, Clustal V, Clustal W, Clustal X, Clustal Ω
Only the last one is used today
Scoring function
First, there should be a scoring function
Clustal uses a simple one. The sum of all v/s all
\[ \begin{aligned} \text{Score}_k(s_{1}[i_1],…,s_{k}[i_k])= & \text{Score}_2(s_{1}[i_1],s_{2}[i_2]) + \\ & \text{Score}_2(s_{1}[i_1],s_{3}[i_3]) + \cdots+ \\ & \text{Score}_2(s_{k-1}[i_{k-1}],s_{k}[i_k]) \end{aligned} \]
where \(\text{Score}_2(s_{a}[i],s_{b}[j])\) is PAM, BLOSUM, or a similar substitution scoring matrix
Progressive alignment order
We start by comparing sequences all-to-all
That is, comparing all pairs of sequences
We store them in a distance matrix
How many pairs can be done with \(N\) sequences?
Building a guide tree
Once we get all pairwise “distances” (that is, scores)
We make a tree by hierarchical clustering or neighbor joining
This is not a phylogenetic tree
The guide tree is built without seeing the big picture
So it is not safe to assign any meaning to it
We will talk more about trees and build phylogenetic trees later
Guide tree guides the alignment
Clustal aligns the sequences following the guide tree
First, it aligns the more similar sequences
Then it adds the nearest sequence, and so on
These are semi-global alignments
Uses \(\text{Score}_k()\) when there are \(k\) sequences